Robert T. Buche

Control of mobile communications with time-varying channels in heavy traffic

Consider the forward link of a system with K remote units and a single base transmitter with time varying connecting channels. Data to be transmitted to the remote units arrives according to some random process, and is queued according to its destination. Power is to be allocated to the K channels in a queue and channel state depend on way to minimize some cost criterion. The channel fading rate is fast and the bandwidth and data arrival rates are high. Owing to the high speed of the fading, arrival, and service rates, an asymptotic or averaging of the heavy traffic type method is promising. By heavy traffic, we mean that on the average there is little server idle time and spare power over the average requirements. Heavy traffic analysis eliminates inessential details and focuses on the fundamental issues of scaling and parametric dependencies. To illustrate the scope of the method, several models are considered, and more complicated systems are also treated.

Control of mobile communication systems with time-varying channels via stability methods

Consider the forward link of a mobile communications system with a single transmitter and connecting to K destinations via randomly varying channels. Data arrives in some random way and is queued according to the K destinations until transmitted. Time is divided into small scheduling intervals. Current systems can estimate the channel (e.g, via pilot signals) and use this information for scheduling. The issues are the allocation of transmitter power and/or time and bandwidth to the various queues in a queue and channel-state dependent way to assure stability and good operation. The decisions are made at the beginning of the scheduling intervals. Stochastic stability methods are used both to assure that the system is stable and to get appropriate allocations, under very weak conditions. The choice of Lyapunov function allows a choice of the effective performance criteria. The resulting controls are readily implementable and allow a range of tradeoffs between current rates and queue lengths. The various extensions allow a large variety of schemes of current interest to be covered. All essential factors are incorporated into a mean rate function, so that the results cover many different systems. Because of the non-Markovian nature of the problem, we use the perturbed Stochastic Lyapunov function method, which is well adapted to such problems. The method is simple and effective.

Rate of Convergence for Constrained Stochastic Approximation Algorithms

There is a large literature on the rate of convergence problem for general unconstrained stochastic approximations. Typically, one centers the iterate

Stochastic approximation and user adaptation in a competitive resource sharing system

theta_n

Heavy traffic control policies for wireless systems with time-varying channels

about the limit point

Adaptive optimization of least-squares tracking algorithms: with applications to adaptive antenna arrays for randomly time-varying mobile communications systems

bartheta

Heavy Traffic Approximations of a Queue with Varying Service Rates and General Arrivals

and then normalizes by dividing by the square root of the step size

Heavy traffic limits in a wireless queueing model with long range dependence

epsilon_n

Heavy Traffic Methods in Wireless Systems: Towards Modeling Heavy Tails and Long Range Dependence

. Then some type of convergence in distribution or weak convergence of Un, the centered and normalized iterate, is proved. For example, one proves that the interpolated process formed by the Un, converges weakly to a stationary Gaussian diffusion, and the variance of the stationary measure is taken to be a measure of the rate of convergence. See the references in [A. Benveniste, M. Metivier, and P. Priouret, Adaptive Algorithms and Stochastic Approximation, Springer-Verlag, Berlin, New York, 1990; L. Gerencer, SIAM J. Control Optim., 30 (1992), pp. 1200--1227; H. J. Kushner and D. S. Clark, Stochastic Approximation for Constrained and Unconstrained Systems, Springer-Verlag, Berlin, New York, 1978; H. J. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer-Verlag, Berlin, New York, 1997; M. T. Wasan, Stochastic Approximation, Cambridge University Press, Cambridge, UK, 1969] for algorithms where the step size either goes to zero or is small and constant. Large deviations provide an alternative approach to the rate of convergence problem [P. Dupuis and H. J. Kushner, SIAM J. Control Optim., 23 (1985), pp. 675--696; P. Dupuis and H. J. Kushner, SIAM J. Control Optim., 27 (1989), pp. 1108--1135; P. Dupuis and H. J. Kushner, Probab. Theory Related Fields, 75 (1987), pp. 223--244; A. P. Korostelev, Stochastic Recurrent Processes, Nauka, Moscow, 1984; H. J. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Springer-Verlag, Berlin, New York, 1997]. When the iterates of the algorithm are constrained to lie in some bounded set, the limit point is frequently on the boundary. With the exception of the large deviations type [P. Dupuis and H. J. Kushner, SIAM J. Control Optim., 23 (1985), pp. 675--696; P. Dupuis and H. J. Kushner, Probab. Theory Related Fields, 75 (1987), pp. 223--244], the rate of convergence literature is essentially confined to the case where the limit point is not on a constraint boundary. nWhen the limit point is on the boundary of the constraint set the usual steps are hard to carry out. In particular, the stability methods which are used to prove tightness of the normalized iterates cannot be carried over in general, and there is the problem of proving tightness of the normalized process and characterizing the limit process. nThis paper develops the necessary techniques and shows that the stationary Gaussian diffusion is replaced by an appropriate stationary reflected linear diffusion, whose variance plays the same role as a measure of the rate of convergence. An application to constrained function minimization under inequality constraints

Power Control for Mobile Communications With Delayed State Information in Heavy Traffic

q^i(x)le 0,ile p